SUMMARY
This discussion focuses on identifying two norms on a vector space that are not equivalent, specifically within the context of infinite-dimensional spaces. The Euclidean norm and the max norm are proposed, but the more relevant example involves the sup norm and the L1 norm on the space C[0,1], which consists of continuous functions on the unit interval. The conclusion emphasizes that norms in finite-dimensional vector spaces are always equivalent, while infinite-dimensional spaces allow for non-equivalent norms.
PREREQUISITES
- Understanding of vector spaces and their properties
- Familiarity with norm definitions and equivalence
- Knowledge of continuous functions and function spaces
- Basic concepts of sup norm and L1 norm
NEXT STEPS
- Research the properties of infinite-dimensional vector spaces
- Study the definitions and applications of the sup norm
- Explore the L1 norm and its significance in functional analysis
- Examine examples of non-equivalent norms in various function spaces
USEFUL FOR
Mathematicians, students of functional analysis, and anyone interested in the properties of norms in vector spaces, particularly in the context of infinite dimensions.